From biomedical to environmental research, a central problem in image analysis is to recognize and locate important effects. An archetypal example is image analysis of the 3D brain volume or the 2D cortical surface, using both anatomical and functional imaging. Examples also abound in 1D functional data (EEG patterns or anatomical neural fibers), 2D images (microscopy) and 2D spatial data (climate maps). These problems share a common data structure in which smooth functions or images are observed repeatedly and aligned on a fine grid. The goal of localization is to identify regions where the signal is strong or where differences exist between conditions or groups of subjects. While there is a rich collection of tools to analyze imaging data, the focus has been mainly on significance testing and controlling error rates under the null hypothesis and has been limited by practical but unrealistic assumptions about the noise field, compromising error control and statistical power. On the other hand, the functional data analysis approach rightly works under the non-zero mean model but ignores the analytical power of smooth random field theory, which has been so successful in image analysis and could enable similar gains for functional data. The main goal of this proposal is to develop new spatial inference methods that directly address the estimation of non-sparse signals and quantification of their spatial uncertainty, in order to increase statistical power, control error rates and obtain appropriately interpretable results. In the previous cycle of this grant, we established methodology for formal error control in peak detection. This renewal develops location uncertainty and detection power for peaks (Aim 1), and moves further to develop confidence bands and spatial confidence regions for the entire signal (Aim 2) and for excursion sets where the signal exceeds a threshold (Aim 3). Methods are proposed to target both the mean (effect magnitude) and the signal-to-noise ratio (standardized mean or effect size), allowing interpretable inference in the presence of spatially non- constant variance, characteristic of neuroimaging data. The proposal offers clear definitions of spatial inference, and supports the methodology with smooth Gaussian random field theory, forgoing the stationarity and zero-mean assumptions. These methods are rigorously validated and used to map the cognitive effects of addictive substance use in the large NIH-funded Adolescent Brain Cognitive Development (ABCD) study. This proposal uniquely brings together ideas from image and functional data analysis to provide more accurate and interpretable spatial localization of important effects in smooth signals and images. The methods developed in this proposal offer more accurate mapping of the brain and other domains, and higher statistical power to identify locations where important effects occur, enhancing scientific understanding and guiding better targeted follow-up studies.